Summary: Systems evolving under the influence of competing two- and three-body interactions are of particular interest in exploring the stability of the equilibrium states of a strongly interacting many-body system. We present a solvable model based on qubit networks, which allows us to investigate the intricate influence of these couplings on the possible asymptotic equilibrium states. We study the asymptotic evolution of finite qubit networks under two- and three-qubit interactions. As representatives of three-qubit interactions we choose controlled unitary interactions (cu-interactions) with one and two control qubits. It is shown that networks with purely three-qubit interactions exhibit different asymptotic dynamics depending on whether we deal with interactions controlled by one or two qubits. However, when we allow three-qubit interactions next to two-qubit interactions, the asymptotics is dictated by two-qubit interactions only. Finally, we prove that the simultaneous presence of two types of three-qubit interactions results in the asymptotic dynamics characteristic for two-qubit cu-interactions.